> > I see no limitations disallowing that interval between 0.999... and 1 > > to be a metric space. It is infinitely sub-dividable and can be > > understood to have its own metric. How this metric would relate to the > > real numbers would require its own study. > > You would agree that R is a complete metric space, yet remove this > portion? That my good sir is a contradiction. > > > When I say decimal, I simply mean any number represent in our common > > base ten positional numbering system. If this seems an overuse of them > > word I would be interested in hearing a better simple description. > > Then, allowing for an infinite number of digits, lim_n->oo 1-10^n, or > 0.9999... is expressible as a decimal. > > You can write a proof of this fairly directly (go on :) ). > > -- > Glen
Hi Glen, I very much appreciate your comments! I have never studied metric spaces and I do not understand what they would entail, so I cannot speak intelligently on them and probably should not have tried to extend them as applicable.
I do believe 0.999... can be expressed as a decimal as long as we allow infinite repeating groups to be standard. The only reason I would disallow this to be considered standard rather than an extension is the asymmetry - being able to express a number but not it's distance from one. Mind you this requires you believe as I do that that 0.999... does not equal 1.
Do you agree there is an infinite set of real numbers that decimal can never hope to express between zero and any value it can? I use only mathematical induction to believe this, noting that for any value decimal can express, between this an zero would lie an infinity.
I have been looking into cauchy sequences and dedekind cuts. I was more aware of the latter than the former. I would be curious to know what about these system needs 0.999... to equal 1. Decimal notation has this same need. As least in the case of decimal notation, I do not hold this need to reflect the true nature of real numbers. Thanks for the areas of study =)
I would be curious to know if you consider my proof to hold any untruths used as justification for my result. I have always found that my inabilites to accept 0.999... as equal to one are met not in terms of my reservations no matter ~how~ clear I make them, instead always in terms of something else. And people wonder why people are 'unwilling to relinquish' their reservations.
Thanks to you and all the folks on this thread who have been curious to know what possible qualities this potential intervening distance could have, I've been running through conceptualizations and will write something up about them to lend context to those who feel they need to understand this piece of the puzzle in order to be able to ever accept that it could exist. Funny we should need to fully understand something before acknowledging its presense! I very much appreciate that.